The pilot reported that he was landing at a private airport after a cross-country flight. During the landing flare, just prior to touchdown, a small deer ran onto the runway toward the airplane. The pilot increased engine power and attempted to climb; however, the deer struck and substantially damaged the left horizontal stabilizer. The pilot reduced engine power and subsequently landed on the runway without further incident. The pilot indicated that he did not experience any preaccident mechanical malfunctions or failures that would have precluded normal operation of the airplane.